For questions involving random variables uniformly distributed on a subset of a measure space. To be used with [probability] or [probability-theory] tag.

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2
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1answer
99 views

All Cdfs have a uniform distribution on [0, 1]?

Consider the following proposition Proposition C Let $Z = F(X)$; where $F$ is the continuous cumulative distribution function of the random variable X, then $Z$ has a uniform distribution on ...
0
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1answer
52 views

Why is $X/\|X\|_2$ uniformly distributed on a unit sphere when X is n-dimensional standard gaussian vector?

In the proving the above, I see that since $X$ is multivariate gaussian then for any orthogonal matrix $Q$ we have that $QX$ is standard multivariate gaussian. Then I somehow reasoned that $Y=X/\|X\|...
1
vote
1answer
87 views

Uniform distribution over the unit disk

Suppose that $U_1$ and $U_2$ are independent, and identically and uniformly distributed over the unit disk, i.e., for $i = 1,2$, $U_i = (X_i, Y_i)$ and the joint density is \begin{equation} f_{(X_i,...
3
votes
2answers
71 views

Pseudorandom Number Generator Using Uniform Random Variable

I am working out of Mathematical Statistics and Data Analysis by John Rice and ran into the following interesting problem I'm having trouble figuring out. Ch 2 (#65) How could random ...
0
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0answers
55 views

find limit distribution by using central limit theorem.

$$x_1,...,x_n \sim \text{uniform (0,1)}$$ $$Y_n=\sum_i^n X_i$$ I want to find limit distribution by using central limit theorem. $E(Y_n)=n/2$ and $V(Y_n)=n/12$ And Moment generating function $M_{...
1
vote
1answer
34 views

Distribution of a Poisson process with uniformly random parameter

Let $X = Unif(2, 4)$ and $Y=Poisson(X)$. My goal is to find $P(Y=n)$, but I always seem to get stuck on some nasty integral. Here's what I've tried: $P(Y=n) =\int_2^4P(Y=n|X=x)*P(X=x)dx = \int_2^4 ...
4
votes
2answers
82 views

Probability problem: length of new segments

I have a line of length $l$. I divide the line in $n$ segments. I do this by choosing $n - 1$ random points (I mean that the $n - 1$ points are uniformly distributed from $0$ to $l$). I want to add a ...
2
votes
1answer
62 views

Two step random experiment: Density of combined uniform and normal distribution

imagine a random experiment, where first some number $u$ is drawn uniformly on $[c-\varepsilon,c+\varepsilon]$ for $c>0$ and $0<\varepsilon<c$. Next, a $N(u,\sigma^2)$-distributed random ...
0
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2answers
34 views

Sum of X-Uniform(0,1) + Y-Uniform(0,2)

I'm trying to find the CDF the sum of $X$ and $Y$ (which are independent). $X$ is uniform distributed over $[0,1]$ and $Y$ over $[0,2]$. I've seen some similar questions which explain the situations ...
0
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3answers
67 views

Generate prime of x decimal digits using bit-oriented prime generator

I've got a question on stackoverflow where somebody asks to generate a random 18-digit prime. Unfortunately, the only prime generator is the one from OpenSSL. This prime generator is however geared ...
-1
votes
1answer
57 views

How do I renormalize these probability distributions?

So I have two random variables, $X_1$ and $X_2$, both uniformly distributed on $[0, 1]$. If $Y = (X_1 + X_2) / 2$, it will also be distributed between 0 and 1, but it won't be uniformly distributed ...
5
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0answers
58 views

How to quantify the “uniformity” of a distribution of holes in a surface

I want to try to quantify if the distribution of holes over a surface is uniform or not. The holes can have any given shape and can be arranged in any way over the surface. Three examples are ...
0
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1answer
34 views

expectation and geometrical probability problem [closed]

A horizontal line of length $5$ units is divided into two parts. If the first part is of length $X$. Find $E[X]$ where $E[\cdot]$ stands for expectation. how to approach this question ? X can take any ...
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0answers
23 views

Finding conditional probability distribution (X|Y) from (Y|X)

$(Y,X)$ has a joint distribution where the marginal of $X$ is a standard normal and $Y|X \sim U \left[|X|-\frac{1}{2},|X|+\frac{1}{2}\right] $ where $U[a,b] $ means uniform in the interval [a,b]. How ...
1
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0answers
33 views

Uniform distribution of points on a cone constrained to a continuous line

I was hanging lights on a Christmas tree yesterday, and thought of a problem , which may have an easy solution - but not one that I can think of off the top of my head. It is posed as follows: ...
0
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0answers
34 views

Uniform Sampling & CDF inverse

I have a probability exam soon, and our prof told us to study the following question: "Describe a procedure for generating independent identically distributed (i.i.d.) samples of a random variable ...
0
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1answer
42 views

Detecting corrupted data in birthdates of a population

I have a population of N birthdates. Let's assume that birthdates are uniformly distributed over the year. I'm concerned that some of these records have been corrupted, for example by someone ...
0
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0answers
46 views

Cramer-Rao Uniform Distribution

If my data, $X_i\sim U(0,\theta)$ is iid. What is the Cramer-Roa lower bound for a variance estimator such as the sample variance? $ S_n= \frac{1}{n} \sum_{i=1}^n (X_i-\bar{X})^2$ I am stuck ...
1
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0answers
36 views

Expected value for number of occurences

Let $W$ be a random word made from letters which are in set $K$ (letters are uniformly distributed in $W$) . Suppose also that $W$ has finite length ($\geq 2$) and size of $K$ is finite ($\geq 2$). ...
2
votes
2answers
86 views

Maximum of martingales

I need to show whether or not the maximum of two martingales is also a martingale. Originally, I thought yes. But supposedly the answer is no. So as a counter-example, let $U_i$ be $iid$ $unif(0,1)$, $...
0
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0answers
29 views

Expectations of order statistics of uniform RVs, via exponential formulation

If $U_1,\ldots, U_n$ are i.i.d. uniform random variables, then I know that the order statistics satisfy $$(U_{(1)},\ldots, U_{(n)}) \overset{d}{=} \left(\frac{X_1}{\sum_{i=1}^{n+1} X_i}, \frac{X_1+...
0
votes
2answers
62 views

Probability of random variable with uniform distribution on an interval

Let a random variable X have a uniform distribution on the interval $[0, 10]$. Find $P(X(X + 10) > 11)$ Since X has a uniform distribution, the pdf of X is $$ f(x)=\left\{\begin{array}{ccl}c&...
0
votes
1answer
21 views

Prove that two random variables are dependent

Given two random variables X and Y where X is uniformly distributed on [-1,1] and Y = X^2, prove that these two random variables are dependent. Of course, it's clear that they are dependent. But, how ...
2
votes
1answer
76 views

$X_1$, $X_2$ i.i.d RVs, $X_1$ is uniformly distributed. Show $E\left(\frac{X_1}{X_1+X_2}\right)=\frac{1}{2}$

Let $X_1$, $X_2$ be two i.i.d. random variables and $X_1$ is uniformly distributed (discrete) on the set $\{1,2,3\}.$ Show that: $$E\left(\frac{X_1}{X_1+X_2}\right)=\frac{1}{2}$$ Can someone give me ...
1
vote
2answers
41 views

Convergence weakly to exponential random variable for this r.v.

Can I get some insight of how to solve this problem? Let $X_1, X_2, X_3, ...$ be i.i.d. copies of uniform random variable on $[0, 1]$. Let $M_n = \text{min}_{1\leq i <j \leq n} |X_i - X_j|$. Show $...
0
votes
2answers
73 views

Uniform Random Variable: Correlation and Independence

Let X be a uniform random variable defined on the interval $(0,1)$. If $Y = 6X^2−6X+1$, compute the correlation of X and Y . Are X and Y independent? Are X and Y uncorrelated? So my work is. $F(X) =...
0
votes
3answers
39 views

Find cumulative distribution function of uniform distribution

Random variable X has uniform distribution on $[0,1] \cup [2,3]$. Find cdf of variable X. I mean i do not know how to treat this on such strange interval.
1
vote
1answer
62 views

Mutual information for a continuous uniform distribution

I'm trying to compute using matlab the mutual information for an $ \infty $-PAM input (the limit of a very dense PAM constellation) for a range of snr and I got stuck. I'm working with a real-valued ...
1
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1answer
77 views

IID Uniform probability problem [closed]

Three students independently attempt to solve a problem. Assume that the times taken by each student to solve the problem are iid according to U(0,30). Find the probability that the student who fi ...
1
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0answers
22 views

Covariance of uniform random and indicator function dependant on it

Define $I = \begin{cases} 1,& \text{if } X\leq a\\ 0,& \text{if } X\gt a \end{cases}$ $X$ is uniform on $[0,1]$. We want to compute $Cov(I,X)$ which involves $E[IX]$. $E[IX] = E[IX|I=...
0
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0answers
22 views

Find probability from Uniform Distribution [duplicate]

Two numbers are selected at random from the interval (0,1). If these numbers are iid and uniformly distributed, find the probability that the three line segments found by breaking interval into three ...
3
votes
1answer
68 views

Collisions with four bullets

This is a follow-up question of Colliding Bullets. I'm interested in a rigorous calculation of a specific aspect of the referred question. We consider four bullets. Once per second a bullet is fired ...
0
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2answers
49 views

The distribution of the sum and difference of independent uniformly distributed variables

Suppose $X$ and $Y$ are independent uniformly distributed on the interval $[-a/2,a/2]$. What is the density function of $Z=X+Y$ and of $Z=X-Y$? I know that it will be the convolution of densities $...
1
vote
1answer
49 views

Uniform Probobility Distribution Word Problem, grocery store checkout and meeting a friend

The amount of time spent waiting in line at a grocery store express checkout varies from 5 minutes to 15 minutes and follows a uniform distribution. Let X be the amount of time spent waiting in line. ...
4
votes
0answers
86 views

$E[Y^2| (X-Y)^+]$ for $X,Y\stackrel{iid}\sim Unif(0,1)$

I am preparing for a test and am unsure of my workings on this exercise: Calculate $E[Y^2| (X-Y)^+]$ for $X,Y\stackrel{iid}\sim Unif(0,1)$ Where $A^+ = \max (A,0)$ My approach was to calculate ...
2
votes
0answers
63 views

Law of large number for a product of uniform iid random variables (stick breaking)

Let $(X_n)_n$, $n = 1, 2, \dots$, be an iid sequence of random variables uniformly distributed on $(0, 1]$. Set $S_0 = 1$ a.s. and, for $n = 1, 2, \dots$, set $S_n = \prod_{k=1}^n X_k$. Compare $S_n$ ...
0
votes
1answer
103 views

Probability of waiting time

Question: At a railroad junction, a car and a truck arrive between 7:15 and 7:30. A train stops the traffic for five minutes from 7:20. What is the probability that the car and truck waited for ...
1
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1answer
80 views

Find the pdf of $Y = g(X)$, where $X$ is a uniform random variable

The question is as follows: Let $X$ be a uniform random variable over $(-1,2)$. Let $g(x) = |x|$. Find the pdf of $Y = g(X)$. And here is my take so far: $$f(x) = \begin{cases} 1/2 & \text{ ...
1
vote
1answer
55 views

Transformation of variables for a non-monotonic function

Question: Let $U \sim \mathrm{Unif}(−α, α)$ follow the uniform distribution on the interval $(−α, α)$ for some parameter $α > 0$ and consider the transformed random variable $X = \sin(U)$. ...
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0answers
28 views

Distribution of function of uniformly random variables

I am sorry if there is no simple answer to this or the answer is completely obvious but I am approaching my wits end here. Probability isn't my forte, nor am I even a mathematician. I am essentially ...
0
votes
1answer
251 views

Uniform Distribution with Independent Random Variables to compute mean of the present value of a bond.

John wants to purchase a bond which will pay him $X$ thousand dollars after two years, where $X$ is equally likely to be any of the numbers in the set $\{0, 1, 2, 3, 4, 5\}$. John believes that the ...
0
votes
1answer
33 views

Uniform Distribution - Change of Variable

I have been stuck on the following question If $X$ has a cumulative distribution $F(x)$, then show $Y = F(X)$ has a uniform distribution with $U(0,1)$. I attempted to solve this problem by first ...
0
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0answers
15 views

Uniformly distributed arrival times, each willing to wait 15 minutes, what is the probability they meet? [duplicate]

Alice and Bob agree to meet for lunch on a certain day at noon. However, neither is known for punctuality. They both arrive independently at uniformly distributed times between noon and 1 pm on that ...
0
votes
3answers
42 views

density of $X^2$ when $X$ has uniform $[-1, 2]$ distribution

Suppose $X$ has uniform $[-1,2]$ distribution. I am trying to find the density of $Z=X^2$. Here is what I have done thus far: Range($Z$)$=[0,4]$. I began computing the distribution of $Z$ for $z \in ...
1
vote
1answer
38 views

When is an improper Riemann integral equal to Lebesgue integral

My original problem is given $X_i\sim^{iid}U[0,1]$, find $$\lim_{n \rightarrow \infty} (X_1X_2 \cdots X_n)^{1/n} = \lim_{n \rightarrow \infty} (\prod_{i=1}^{n} X_i)^{1/n}$$ Well, $$\lim_{n \...
1
vote
1answer
51 views

Uniform distribution on $[0,1]$ and random variable $Y=\frac{U}{e^{1-U}}$

$U$~$Unif[0,1]$ and we have the random variable $Y=\frac{U}{e^{1-U}}$. Find the density function of $Y$. So far I have that $0\le Y\le1$ and that... $$F_Y(t)=P(\frac{U}{e^{1-U}}\le t)=P(\ln (\frac{U}...
0
votes
0answers
204 views

p-value of uniformity of given distributions,Matlab

Given a vector of real numbers $[a_0,...,a_n]$, how do I find the $p$-value (in Matlab, say) that it is drawn from the uniform distribution over [0,1]? I.e. $H_0$ is the hypotheses that it is drawn ...
1
vote
2answers
74 views

Probability Xavier and Yolanda meet for lunch

Xavier and Yolanda plan to meet for lunch between noon and 1 p.m. They arrive independently with uniform distribution on [0, 1]. Yolanda will wait 30 min. for Xavier, but Xavier will only wait 15 min. ...
0
votes
1answer
34 views

$P\left(X+\frac{10}{X}>7\right)$ of a uniform distribution

Problem: $X$ has a continuous uniform distribution on $[0,10]$. Find $P\left( X + \frac{ 10 } { X } >7\right)$. So far, I have the PDF $f(x) = 1/10$ and CDF $F(x) = x/10$ for $0 < x < 10$. ...
1
vote
1answer
35 views

Density function of uniform prob distribution

Let $X ∼\operatorname{Uniform}(0,1)$. Find the density function of $Y = e^X$. I got to: $F_Y(y)$=$P(Y\le y)$=$P(e^X\le y)$=$P(X\le \ln(y))$ Not sure where to go from here?